Let a, b, d be non-negative integers. A graph G is (d, a, b)∗ -colorable if its vertex set can be partitioned into a + b sets I1,...,Ia,O1,...,Ob such that the graph G[Ii] induced by Ii has maximum degree at most d for 1 ≤ i ≤ a, while the graph G[Oi] induced by Oi is an edgeless graph for 1 ≤ i ≤ b. In this paper, we give two real-valued functions f and g such that any graph with maximum average degree at most f(d, a, b) is (d, a, b)∗ -colorable, and there exist non- (d, a, b)∗ -colorable graphs with average degree at most g(d, a, b). Both these functions converge (by lower values) to 2a + b when d tends to infinity. Counterintuitively, this implies that allowing a color to be of type Ii even for a large degree d increases the maximum average degree that guarantees the existence of a valid coloring only by 1. Using a color of type Ii (even with a very large degree d) is somehow less powerful than using two colors of type Oi (two stable sets).